Related papers: An Enumerative Geometry for Magic and Magilatin La…
This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…
For a set of distances $D$, a graph $G$ of order $n$ is said to be $D-$magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots, n\}$ and a constant $k$ such that for any vertex $x$, $\sum_{y\in N_D(x)} f(y) =k$, where…
In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…
We prove that the inverse of a positive-definite matrix can be approximated by a weighted-sum of a small number of matrix exponentials. Combining this with a previous result [OSV12], we establish an equivalence between matrix inversion and…
Bar Codes are combinatorial objects encoding many properties of monomial ideals. In this paper we employ these objects to study Janet-like divisions. Given a finite set of terms U, from its Bar Code we can compute the Janet-like…
An integer $n$ is said to be \textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an…
We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by north-east lattice paths going from the origin to a point (k,n). The numbers from 1 and n…
A procedure for the construction and the classification of multilattices in arbitrary dimension is proposed. The algorithm allows to determine explicitly the location of the points of a multilattice given its space group, and to determine…
An antimagic labelling of a graph $G = (V,E)$ is a bijection from $E$ to $\{1,2, \ldots, |E|\}$, such that all vertex-sums are pairwise distinct, where the vertex-sum of each vertex is the sum of labels over edges incident to this vertex. A…
A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error…
Let $f$ be a polynomial system consisting of $n$ polynomials $f_1,\cdots, f_n$ in $n$ variables $x_1,\cdots, x_n$, with coefficients in $\mathbb{Q}$ and let $\langle f\rangle$ be the ideal generated by $f$. Such a polynomial system, which…
For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the…
The aim of this note is to introduce fastest new general methods for the construction of double and single even order magic squares. As in [5], the method for double even order magic squares is fairly straight-forward but some adjustments…
We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the…
We consider graph labelings with an assignment of odd prime numbers to the vertices. Similarly to graceful graphs, a labeling is said to be elegant if the absolute differences between the labels of adjacent vertices describe exactly the…
Let $\mathbb{Q}$ (resp. $\mathbb{R}$) be the field of rational (resp. real) numbers and $X = (X_1, \ldots, X_n)$ be variables. Deciding the non-negativity of polynomials in $\mathbb{Q}[X]$ over $\mathbb{R}^n$ or over semi-algebraic domains…
Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of…
The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The…
A "numerical set-expression" is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of…
We study the correspondence between unicritical laminations and maximally critical laminations with rotational and identity return polygons. Laminations are a combinatorial and topological way to study Julia sets. Laminations give…